Nuprl Lemma : real-continuity2-ext

`∀a,b:ℝ.`
`  ∀f:[a, b] ⟶ℝ`
`    ((∀x,y:{x:ℝ| x ∈ [a, b]} .  (f x ≠ f y `` x ≠ y))`
`    `` (∀k:ℕ+. ∃d:{d:ℝ| r0 < d} . ∀x,y:{x:ℝ| x ∈ [a, b]} .  ((|x - y| ≤ d) `` (|(f x) - f y| ≤ (r1/r(k)))))) `
`  supposing a ≤ b`

Proof

Definitions occuring in Statement :  rfun: `I ⟶ℝ` rccint: `[l, u]` i-member: `r ∈ I` rdiv: `(x/y)` rneq: `x ≠ y` rleq: `x ≤ y` rless: `x < y` rabs: `|x|` rsub: `x - y` int-to-real: `r(n)` real: `ℝ` nat_plus: `ℕ+` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` apply: `f a` natural_number: `\$n`
Definitions unfolded in proof :  real-continuity-ext real-continuity2 ifthenelse: `if b then t else f fi ` bfalse: `ff` it: `⋅` btrue: `tt` subtract: `n - m` member: `t ∈ T`
Lemmas referenced :  real-continuity2 real-continuity-ext
Rules used in proof :  equalitySymmetry equalityTransitivity sqequalHypSubstitution thin sqequalRule hypothesis extract_by_obid instantiate cut sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution introduction

Latex:
\mforall{}a,b:\mBbbR{}.
\mforall{}f:[a,  b]  {}\mrightarrow{}\mBbbR{}
((\mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  [a,  b]\}  .    (f  x  \mneq{}  f  y  {}\mRightarrow{}  x  \mneq{}  y))
{}\mRightarrow{}  (\mforall{}k:\mBbbN{}\msupplus{}
\mexists{}d:\{d:\mBbbR{}|  r0  <  d\}
\mforall{}x,y:\{x:\mBbbR{}|  x  \mmember{}  [a,  b]\}  .    ((|x  -  y|  \mleq{}  d)  {}\mRightarrow{}  (|(f  x)  -  f  y|  \mleq{}  (r1/r(k))))))
supposing  a  \mleq{}  b

Date html generated: 2018_05_22-PM-02_11_39
Last ObjectModification: 2018_05_21-AM-00_27_29

Theory : reals

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